We introduce and discuss certain kinetic models of (continuous) opinion formation involving both exchange of opinion between individual agents and diffusion of information. We show conditions which ensure that the kinetic model reaches non trivial stationary states in case of lack of diffusion in correspondence of some opinion point. Analytical results are then obtained by considering a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of opinion among individuals.

We analyse the uniqueness of the solutions of a PDE system from the framework of compressible turbulent models. Smoothness properties of the turbulent viscosity closure law are of central importance. Several closures of practical importance, including the most widely used law, indeed fail to be Lipschitz continuous in the natural neighborhood of a null turbulent energy. For such models, we prove the existence of infinitely many distinct traveling wave solutions which exhibit positive turbulent energy but connect at infinity end states with vanishing turbulence. Examples and counter-examples are given.

We introduce a new approach to deal with the numerical solution of partial differential equations on surfaces. Thereby we reformulate the problem on a larger domain in one higher dimension and introduce a diffuse interface region of a phase-field variable, which is defined in the whole domain. The surface of interest is now only implicitly given by the $1=2$-level set of this phase-field variable. Formal matched asymptotics show the convergence of the reformulated problem to the original PDE on the surface, as the diffuse interface width shrinks to zero. The main advantage of the approach is the possibility to formulate the problem on a Cartesian grid. With adaptive grid refinement the additional computational cost resulting from the higher dimension can be significantly reduced. Examples on linear diffusion and nonlinear phase separation demonstrate the wide applicability of the method.

We consider a multi-class (resp. source destination) model for traffic flow as introduced in M. Garavello and B. Piccoli, Source-Destination flow on a road network. We propose a reformulation and discuss the extension of this model to road networks by proposing criteria for solving Riemann problems at a road intersection. We present some numerical simulations for the derived conditions.

In this paper we present a global existence and uniqueness result of solutions to a system consisting of a coupled nonlinear PDE and a nonlinear ODE. This system is a one-dimensional fluiddynamic model for tunnel fires and was presented in I. Gasser and J. Struckmeier and then analyzed in I. Gasser and in I. Gasser and H. Steinrück with respect to the applications

Coupled quantum-fluid models are derived by means of a diffusion approximation from adiabatic quantum-kinetic models. These models describe the electron transport of a bidimensional electron gas. Particles are confined in one direction (denoted by $z$) while transport occurs in an orthogonal direction (denoted by $x$). The length-scale in the $z$ direction is comparable to the de Broglie wavelength, while the $x$-length scale is much bigger. The aim of this paper is to investigate the diffusion limit from quantum-kinetic to quantum-fluid models, which are numerically more interesting. Transitions between sub-bands are considered in the Fermi Golden rule setting.

A numerical technique for the reconstruction of diffusion processes (diffusions, in short) from data is presented. The drift and diffusion coeffcients of the generator of the diffusion are found by minimizing an object function which measures the difference between the eigenspectrum of the operator and a reference eigenspectrum. The reference spectrum can be obtained, in discretized form, from time-series through the construction of a discrete-time Markov chain. Discretization of the Fokker-Planck operator turns minimization of the object function into a quadratic programming problem on a convex domain, for which well-established solution methods exist. The technique is a generalization of a reconstruction procedure for continuous-time Markov chain generators, recently developed by the authors. The technique also allows us to derive the coeffcients in the homogenized diffusion for the slow variables in systems with multiple timescales.

The instability of variable media to a broad class of long waves having dispersion relations that are an odd function of wavenumber is examined. For Hamiltonian media, new necessary conditions for the existence and structure of global modes are obtained. For non-Hamiltonian media, an analysis of the complex WKB branch points yields explicit expressions for the frequency and structure of the global modes, which manifest as spatially oscillatory wave packets or smooth envelope structures. These distinct modes and their locations within the media can be predicted by simply examining the local convergence or divergence of the group velocity in the long wave limit.